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Cuchulainn
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Re: About solving a transport equation

February 13th, 2020, 12:14 pm

1) You like transforming coords. So change to coords that move with the chars.

Or

2) Constrain dy and dt so that you solve numerically along the chars. That’s what I’ve used in the past.
1) Don't understand that comment. I just used several standard fd schemes directly on Test Case 2.
2) That is possible; however, it is not easy for me and it is difficult to generalise, seems to be the general consensus. 
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Cuchulainn
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Re: About solving a transport equation

February 13th, 2020, 1:04 pm

A good next example is the inviscid (and possibly viscous) Burgers' equation using the methods discussed to date.

Test Case 3

[$]\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0.[$] ... advective form (3A)

or 

[$]\frac{\partial u}{\partial t} + \frac {1}{2}\frac{\partial u^2}{\partial x} = 0.[$] .. conservative form (3B)

Eventually, the viscous PDE

[$]\frac{\partial u}{\partial t} + \frac {1}{2}\frac{\partial u^2}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2.} [$]  (3C)


https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

[$]u(x,t) = (ax + b)/(at + 1)[$]

for the initial condition

[$]u(x,0) = ax + b.[$]
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FaridMoussaoui
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Re: About solving a transport equation

February 13th, 2020, 1:30 pm

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.
 
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Alan
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Re: About solving a transport equation

February 13th, 2020, 2:34 pm



https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

[$]u(x,t) = (ax + b)/(at + 1)[$]

for the initial condition

[$]u(x,0) = ax + b.[$]
I said earlier I don't know much about shocks. But, reading that Wikipedia link, it's trivial to start with their "implicit relation" [$]u = f(x - u \, t)[$], and linear [$]f[$], and then find the given solution for times such that the denominator does not vanish: [$]t < -1/a[$]. Presumably from the physics, [$]a < 0[$] is the interesting case.

It's also easy to see that, if u solves the general implicit relation (and f has a derivative), then you get [$](u_t + u u_x)(1 + f' t) =0[$]. So, unless [$]1 + f' t = 0[$], you get a general PDE problem solution. The vanishing of the second factor must lead to Wikipedia's formula for their "breaking time" [$]t_b[$], but needs some thought. Their formula for that does check out for the linear case example.
 
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Cuchulainn
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Re: About solving a transport equation

February 13th, 2020, 4:24 pm

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.
That's not the main goal, really. See it more as a benchmark case as it were. We can test many methods on it.
But we want to test Roe et al (mainly Laney) against other schemes. I am more interested in my own discoveries etc, And I am sure Alan and Paul will have fresh viewpoints.

BTW Farid, which methods do you recommend for (3A), (3B)?

Most of the books don't have the detail.
Last edited by Cuchulainn on February 13th, 2020, 4:49 pm, edited 2 times in total.
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Paul
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Re: About solving a transport equation

February 13th, 2020, 4:30 pm

I don't have any fresh viewpoints! I'm just regurgitating some basic things I learned 40 years ago!
 
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Cuchulainn
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Re: About solving a transport equation

February 13th, 2020, 6:55 pm

I don't have any fresh viewpoints! I'm just regurgitating some basic things I learned 40 years ago!
Total Recall?
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Cuchulainn
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Re: About solving a transport equation

February 13th, 2020, 7:06 pm



https://en.wikipedia.org/wiki/Burgers%27_equation

It would be interesting if Alan and Paul can produce explicit solution for (3A)/(3B)

[$]u(x,t) = (ax + b)/(at + 1)[$]

for the initial condition

[$]u(x,0) = ax + b.[$]
I said earlier I don't know much about shocks. But, reading that Wikipedia link, it's trivial to start with their "implicit relation" [$]u = f(x - u \, t)[$], and linear [$]f[$], and then find the given solution for times such that the denominator does not vanish: [$]t < -1/a[$]. Presumably from the physics, [$]a < 0[$] is the interesting case.

It's also easy to see that, if u solves the general implicit relation (and f has a derivative), then you get [$](u_t + u u_x)(1 + f' t) =0[$]. So, unless [$]1 + f' t = 0[$], you get a general PDE problem solution. The vanishing of the second factor must lead to Wikipedia's formula for their "breaking time" [$]t_b[$], but needs some thought. Their formula for that does check out for the linear case example.
Burgers' etc. is new for me too (Paul and Farid have a 40 year headstart). 

This looks like a nice overview.
https://www.iist.ac.in/sites/default/files/people/Burgers_equation_inviscid.pdf
Step over the gap, not into it. Watch the space between platform and train.
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http://www.datasim.nl
 
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Paul
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Re: About solving a transport equation

February 13th, 2020, 7:15 pm

Cat [$]\in[$] pigeons:

Jump conditions

Weak solutions

Multiply 3(A) by any power of u and write in conservation form
 
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FaridMoussaoui
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Re: About solving a transport equation

February 13th, 2020, 8:21 pm

BTW Farid, which methods do you recommend for (3A), (3B)?
use any second-oder Riemann solver.
 
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FaridMoussaoui
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Re: About solving a transport equation

February 13th, 2020, 9:26 pm

There is a software out there called CLAWPACK. Under the hood, it is classical fortran routines supplied with python interface.

Have a look to the gallery of test cases.

As usual the installation on Unix-like systems (Linux/OSX) is seamless. They said nothing about Windows OS but if you install Cygwin it should work.
When I switch to windows OS, I use it a lot (cygwin) as I like working with command line. 
 
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Cuchulainn
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Re: About solving a transport equation

February 15th, 2020, 10:05 am

OK, I installed Cygwin and will take that route now.
My main interest is finding out how and why these numerical methods work. Maybe it's called reverse engineering/osmosis.

I am looking at approximate Riemann solvers, In fact, my upwind scheme is similar to Roe's method since it is essentially a linearisation of a quasilinear pde. 
BTW Tannehill et al book is a nice overview of some of these methods. 
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FaridMoussaoui
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Re: About solving a transport equation

February 15th, 2020, 2:04 pm

A good book for this kind of solvers is Toro  "Riemann Solvers and Numerical Methods for Fluid Dynamics"

A good intro paper is "Wave Propagation Algorithms for Multidimensional Hyperbolic Systems"

If you can't access freely the paper, I can send you a copy.
 
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Cuchulainn
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Re: About solving a transport equation

February 15th, 2020, 3:26 pm

Thx!
Step over the gap, not into it. Watch the space between platform and train.
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JohnLeM
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Re: About solving a transport equation

February 17th, 2020, 11:10 am

Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.
That's not the main goal, really. See it more as a benchmark case as it were. We can test many methods on it.
But we want to test Roe et al (mainly Laney) against other schemes. I am more interested in my own discoveries etc, And I am sure Alan and Paul will have fresh viewpoints.

BTW Farid, which methods do you recommend for (3A), (3B)?

Most of the books don't have the detail.
Trying to be useful there, I just mention that we proposed some years ago  in this paper a method to compute explicitly solutions to these equations : more precisely one can compute explicitly (entropic or conservative) multi-dimensional solutions for conservation laws or Hamilton-Jacobi type equations, with convex fluxes (of Burger type) or non convex fluxes (to model multi-phase medium).
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